Thanks for the suggestion, but even after adding a share as \\hostname\SQLBACKUP\database, I'm still getting this error:

Msg 3153, Level 16, State 2, Line 1

The database is already fully recovered.

Msg 3013, Level 16, State 1, Line 1

RESTORE DATABASE is terminating abnormally.

As soon as I change the script to point to C:\SQL_Backup\Database\ it immediately restores.

The databases are fully functional and aren't restoring any logs, so the error message is confusing.